Classical Lamination Theory

A reference guide for engineers and students working with composite laminates.

What is Classical Lamination Theory?

Classical Lamination Theory (CLT) is the standard analytical framework for predicting the mechanical response of fiber-reinforced composite laminates¹. It converts the properties of individual plies (lamina), obtained either from test data or from micromechanics models, into laminate-level stiffness and compliance matrices, enabling engineers to predict how a laminate will deform under combined in-plane loads and bending moments.

This page is a practical reference, not a full derivation. There are excellent textbooks¹³ that cover the theory in detail. If you are using ABD Composites, you likely already know the basics. The goal here is to give you a quick refresher on the key concepts and clarify what each output of the calculator means.

CLT has been the foundation of composite structural analysis since the 1960s. It is based on three key assumptions:

• Thin plate assumption: The laminate thickness is small compared to its in-plane dimensions, following Kirchhoff-Love plate theory².
• Plane stress: Out-of-plane stress components (σ₃, τ₁₃, τ₂₃) are negligible. Valid for thin laminates under in-plane or bending loads.
• Perfect bonding: No slip between plies. Strains are continuous across the thickness.

Coordinate Systems

Two coordinate systems are used in CLT¹:

Each ply has an orientation angle θ (theta) measured from the laminate x-axis to the fiber direction (1-axis), following the right-hand rule. This angle determines the transformation matrix used to rotate ply properties into the laminate frame.

Material coordinates (1-2-3): Aligned with the ply's fiber direction.

• 1: Fiber direction (highest stiffness)
• 2: Transverse to fibers (in-plane)
• 3: Through-thickness (normal to ply plane)

Laminate coordinates (x-y-z): Fixed to the laminate reference plane.

• x, y: In-plane directions
• z: Through-thickness, positive toward the top ply

The z-axis origin is the midplane, so the bottom surface of the laminate is at z = -h/2 and the top surface at z = +h/2. Every through-thickness result on this site (the B matrix, curvatures, ply z-coordinates and the sense of an applied moment) is reported in that frame.

The ABD Matrix

The central result of CLT is the ABD matrix¹: a 6×6 constitutive relation that links the applied force and moment resultants {N, M} to the midplane strains and curvatures {ε⁰, κ}:

The three submatrices have distinct physical meanings:

• [A], extensional stiffness: Relates in-plane forces to midplane strains. Aᵢⱼ = Σ Q̄ᵢⱼ(k) · (zk - z(k-1)) where the sum runs over all plies.
• [B], coupling stiffness: Couples in-plane forces to curvatures (and moments to midplane strains). B = 0 for symmetric laminates.
• [D], bending stiffness: Relates moments to curvatures. Dᵢⱼ = 1/3 Σ Q̄ᵢⱼ(k) · (zk³ - z(k-1)³).

Q̄ᵢⱼ(k) is the off-axis reduced stiffness matrix of ply k, obtained by rotating the ply's on-axis stiffness matrix Q using the transformation matrix T(θ). The on-axis Q matrix is built from the ply's elastic constants (E₁, E₂, G₁₂, v₁₂), which can be measured experimentally or predicted from fiber and matrix data using micromechanics.

Why 6x6?

You might notice the subscripts in the ABD matrix only use 1, 2, and 6. These are Voigt notation for the three in-plane stress components: 1 = xx, 2 = yy, 6 = xy. The "missing" indices 3, 4, and 5 correspond to through-thickness normal stress (zz) and the two transverse shear stresses (yz, xz). The plane stress assumption sets all three to zero, so they simply do not appear. There is no A₁₃ term because there is no index 3 in the formulation.

The 6x6 ABD matrix covers the vast majority of preliminary laminate design work. When more detailed predictions are required, engineers turn to theories that include the through-thickness stresses: First-Order Shear Deformation Theory (FSDT) for thick laminates where transverse shear matters, or full 3D finite element analysis for stress states near free edges, holes, bolted joints, and ply drops where through-thickness interlaminar stresses drive delamination.

Individual Terms of the A Matrix

Each term in the [A] matrix controls a specific in-plane response:

• A₁₁, A₂₂: Extensional stiffness in the x and y directions. How much force is needed to stretch the laminate in each direction.
• A₁₂: Poisson coupling between x and y. Pulling the laminate in x causes contraction in y, and vice versa.
• A₆₆: In-plane shear stiffness. Resistance to shearing deformation in the x-y plane.
• A₁₆, A₂₆: Extension-shear coupling. When nonzero, pulling the laminate in x or y also produces shear deformation. This is the primary reason engineers use balanced laminates: pairing every +θ ply with a -θ ply zeroes both A₁₆ and A₂₆, preventing unexpected shear under normal loading.

Individual Terms of the B Matrix

The [B] matrix couples in-plane and bending behavior. All B terms are zero for symmetric laminates, which is the main reason symmetry is preferred in practice.

• B₁₁, B₂₂: Extension-bending coupling. Pulling in x or y causes the laminate to bend, and applying a bending moment causes stretching. This is the source of the warping problem in asymmetric laminates during cool-down from cure temperature.
• B₁₂: Cross-coupling between in-plane forces and out-of-plane curvatures in the orthogonal direction.
• B₁₆, B₂₆: In-plane normal forces cause twisting, and bending moments cause in-plane shear. Among the most counterintuitive coupling terms in CLT.
• B₆₆: In-plane shear force causes twisting curvature, and twisting moment causes in-plane shear strain.

Individual Terms of the D Matrix

Each term in the [D] matrix controls a specific bending or twisting response. Unlike A terms, D terms depend heavily on stacking order because plies far from the midplane contribute disproportionately (cubic z-weighting).

• D₁₁, D₂₂: Bending stiffness in x and y. Moving stiff plies (e.g. 0° plies) to the outer surfaces increases D₁₁ significantly without changing A₁₁.
• D₁₂: Poisson coupling in bending. Bending the laminate about one axis induces curvature about the other axis.
• D₆₆: Twisting stiffness. Resistance to twisting deformation of the laminate.
• D₁₆, D₂₆: Bend-twist coupling. When nonzero, bending the laminate also causes it to twist. This effect is used deliberately in aeroelastic tailoring of aircraft wing skins, but is generally unwanted in other structures. Exactly zero for cross-ply laminates. For balanced-symmetric laminates they are not strictly zero, but tend toward zero as the number of plies increases and the +θ/-θ pairs are interleaved rather than grouped.

The ABD matrix as computed in the dashboard. The A, B and D blocks are shown separately, so a zero B block is visible at a glance.

Engineering Constants

The effective engineering constants of the laminate are derived from the compliance matrix (the inverse of the ABD matrix). These are "smeared" properties that represent the equivalent behavior of the laminate as a whole, useful for preliminary structural sizing and comparison between layup alternatives. This section covers the idea; the full symbol-by-symbol lookup, including the restrained-contraction moduli and the non-dimensional plate parameters, is on the Engineering Constants reference page.

Membrane (In-Plane) Constants

The in-plane compliance sub-matrix a yields the membrane engineering constants. For symmetric laminates (B = 0), a = A^-1 directly. For asymmetric laminates, a is the top-left 3×3 block of the full ABD^-1 matrix, not simply A^-1.

• E_x: Effective Young's modulus in the x-direction
• E_y: Effective Young's modulus in the y-direction
• G_xy: Effective in-plane shear modulus
• ν_xy: Major Poisson's ratio, contraction in y when the laminate is loaded in x
• ν_yx: Minor Poisson's ratio, contraction in x when the laminate is loaded in y

The two Poisson's ratios are generally different for an anisotropic laminate, but they are related by the reciprocal relation ν_xy/E_x = ν_yx/E_y (analogous to the ply-level relation ν₁₂/E₁ = ν₂₁/E₂). A laminate that is stiffer in x than y will have a larger ν_xy than ν_yx.

Tensile-Shear Coupling

For general (unbalanced) laminates, two additional dimensionless constants describe how normal and shear responses are coupled:

• η_xy,x: Shear strain per unit normal strain under uniaxial loading in x. When nonzero, pulling the laminate along x also skews it in shear.
• η_xy,y: Shear strain per unit normal strain under uniaxial loading in y.

For a balanced laminate (every +θ ply paired with a -θ ply), both coupling coefficients are zero. This is one of the primary reasons balanced laminates are preferred in practice: the laminate deforms predictably without parasitic shear under normal loading.

Flexural (Bending) Constants

The bending compliance sub-matrix d yields the flexural engineering constants. For symmetric laminates (B = 0), d = D^-1 directly. For asymmetric laminates, d is the bottom-right 3×3 block of ABD^-1.

• E_x^f: Flexural modulus in x (bending stiffness equivalent)
• E_y^f: Flexural modulus in y
• G_xy^f: Flexural shear modulus (twisting stiffness equivalent)
• ν_xy^f: Flexural Poisson's ratio

Flexural constants differ from membrane constants because plies far from the midplane contribute disproportionately to bending stiffness (cubic weighting in D), while all plies contribute equally per unit thickness to membrane stiffness (linear weighting in A). This means a laminate's bending modulus depends on the stacking order, even though its membrane modulus does not. These constants are particularly important for bending-dominated structures like plates and shells.

Effective Thermal Expansion (CTE)

When a laminate is subjected to a uniform temperature change, the effective coefficients of thermal expansion describe the resulting free strains at the laminate level. The ply-level CTEs (CTE₁, CTE₂) that feed into this calculation can be measured or predicted from micromechanics thermal models:

• CTE_x, CTE_y: Membrane CTE in x and y. The laminate-level expansion per degree, accounting for the constraining effect of plies at different orientations.
• CTE_xy: Thermal shear coefficient. Zero for balanced laminates.
• κ_x^T, κ_y^T, κ_xy^T: Thermal curvature coefficients. For asymmetric laminates (B ≠ 0), a temperature change also causes warping. Zero for symmetric laminates.

Effective Moisture Expansion (CME)

Moisture absorption produces swelling strains analogous to thermal expansion:

• CME_x, CME_y: Membrane CME in x and y. The laminate-level swelling per unit moisture concentration change.
• CME_xy: Moisture shear coefficient. Zero for balanced laminates.
• κ_x^M, κ_y^M, κ_xy^M: Moisture curvature coefficients. For asymmetric laminates, moisture gradients can cause warping. Zero for symmetric laminates.

Symmetric Laminates

A laminate is symmetric when the ply sequence is mirrored about its midplane. For example, [0/45/90]s (the s denotes symmetric) expands to [0/45/90/90/45/0].

Symmetric laminates have [B] = 0, which means there is no extension-bending coupling. This simplifies the analysis considerably: in-plane loads cause only in-plane deformation, and bending moments cause only curvature. Symmetric laminates also avoid warping during cooling from cure temperature, making them easier to manufacture.

Most practical structural laminates are designed to be symmetric for these reasons.

Laminate Classification

Laminates are classified based on their stacking sequence and the coupling behavior that results³. These classifications have direct consequences for analysis, manufacturing, and structural performance. The ABD Composites calculator automatically detects and displays these as classification badges.

A cross-ply laminate contains only 0° and 90° plies. This zeroes out A₁₆, A₂₆, D₁₆, and D₂₆, eliminating both extension-shear and bend-twist coupling. Cross-ply laminates are inherently balanced since 0° and 90° plies are self-balancing.

A well-designed structural laminate is typically at least symmetric and balanced. The calculator automatically detects and displays these as chips:

Look for these chips in the free calculator and the dashboard laminate builder as you adjust your layup.

Laminate Notation

Laminate stacking sequences are written using standard shorthand notation¹. Angles are listed in stacking order, one ply per entry. The laminate builder in the ABD Composites dashboard automatically generates the laminate code from your stacking sequence using the notation described below.

• [0/45/-45/90]: general non-symmetric 4-ply laminate. Angles are listed in stacking order separated by slashes.
• [0/±45/90]_s: symmetric 8-ply laminate. The subscript s indicates the sequence is mirrored about the midplane, producing [0/45/-45/90/90/-45/45/0]. The ± notation means a +45° ply immediately followed by a -45° ply. This only applies when the positive angle is immediately followed by the corresponding negative angle.
• [0₂/±45/90]_s: a subscript integer means consecutive identical plies. 0₂ = two 0° plies in a row. This produces a 10-ply symmetric laminate: [0/0/45/-45/90/90/-45/45/0/0].
• [0/45/90]_s: the overline marks the midplane ply in an odd-count symmetric laminate. This 5-ply laminate expands to [0/45/90/45/0]. The overlined 90° ply sits on the midplane and is not doubled. Without the overline a 3-input symmetric laminate would incorrectly double the last ply.

The ± notation applies only when a positive angle θ is immediately followed by -θ in the stacking sequence. It does not apply to 0° or 90° plies. If two ±45 pairs appear in a row they collapse further: [±45/±45] becomes [±45₂].

Which end of the layup code is the bottom ply?

A layup code on its own does not say which end of the list is the bottom of the laminate. [0/90/45] can mean the 0° ply is the top surface and the 45° ply the bottom, or the other way round. Both readings are in common use, and a printed code rarely states which one it follows.

The two readings are the same plies in the opposite through-thickness order, so they describe genuinely different laminates unless the layup is symmetric. What changes:

• Every term of the B matrix changes sign. B integrates z, which is odd about the midplane, so turning the laminate over negates it. The extensional-bending coupling is the same magnitude and the opposite sense.
• The coupling response reverses. The curvature an in-plane load produces, and the mid-plane strain a moment produces, both change sign. Turning the laminate over is equivalent to reversing the applied moment. Curvature under a pure moment is unchanged, because D is unchanged.
• Per-ply stresses swap ends under bending. The ply that was outermost in tension is now outermost in compression, which can change which ply fails first. Under a pure in-plane load they are unchanged, ply for ply, because turning the laminate over only mirrors where each ply sits.

And what does not change:

• A and D are identical, to the last digit. Both integrate even powers of z, so they cannot tell the two readings apart. In-plane stiffness and bending stiffness are unaffected.
• A symmetric laminate is unaffected entirely. Its stacking sequence is a palindrome, so it reads the same from either end and B is zero in both.

Both readings appear in the literature, so ABD Composites fixes one and uses it everywhere: the code reads top-to-bottom. The first element of [0/90/45] is the top ply, the last element is the bottom ply, and ply 1 is the top ply, in the layup builder, the layup code, every result table and every chart. The layer list is drawn in that same order, TOP above the rows and BOTTOM below, and that order is what is saved, exported and copied, so the same laminate is the same physical stack for everyone who opens it. The free calculator on this site reads a code the same way. The frame is right-handed with z positive upward: the top ply is the ply at z = +h/2.

To build the turned-over laminate, which per the list above is a genuinely different laminate, use Reverse stack in the builder: A and D stay put and every B term changes sign. A symmetric layup is the happy exception to all of this: its stack is a palindrome, so its code is the same string read from either face and reversing it changes nothing.

Author: Rick Schrijver

References

1. Jones, R.M. Mechanics of Composite Materials, 2nd ed., Taylor & Francis, 1999.
2. Kirchhoff, G.R. "Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe," Journal fur die reine und angewandte Mathematik, vol. 40, pp. 51-88, 1850. doi:10.1515/crll.1850.40.51
3. Tsai, S.W. and Hahn, H.T. Introduction to Composite Materials, Technomic Publishing, 1980.

Frequently Asked Questions